On a ranking point system, I need the sum of each rank to equal 100 and to gradually increase over 13 people?

get ranking
David asked:

Here is my problem:

There are 13 people who share a commission off of one big revenue pool.

Based on their performance, they are ranked #1 through #13.

Each of them takes a percentage of the revenue based on their ranking (#1 gets the biggest percentage, #13 gets the smallest percentage).

I want #1 to get 15%.
I want #13 to get 1%.

Is there a mathematical formula to figure out what percentage I can I can give to #2 through #12 in which the sum of all the percentages, including the 1% and 15% equals 100%?

It’s easy to get the numbers to distribute evenly from 1% to 15% using a contant rate of increase of 1.6666666% for each consecutive ranking but the sum of all the percentages equals 104%. Is there a parabolic way to figure out how to distribute the percentages?



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One Response to 'On a ranking point system, I need the sum of each rank to equal 100 and to gradually increase over 13 people?'

  1. simbha_07 - December 14th, 2008 at 6:29 am

    It’s impossible to have the numbers be evenly spaced, have the sum equal 100 and have the first one be 1 and the last one be 15. You can see this as follows:

    Call the first number 1.
    The next number is 1+a
    The next number is 1+2a.
    etc so the 13th number is 1+12a.

    The sum of these (check it yourself) is 13 + 78a.

    This has to equal 100. That makes a = 87/78 which is approximately 1.115. That’s a unique solution for a.

    However, if one does this the last number will be approximately 14.385.

    So, you’ll have to choose two of the three “constraints”. Assuming you do want to start off with 1 as the “13th ranked commission” then you can go with the above. An alternative is to instead allow each of the first twelve to be 1% higher than the previous and then set the last so the total is 100, as follows:

    (1,2,3,4,5,6,7,8,9,10,11,12,22)

    So the top salesperson will get 22%.

    Another variation that isn’t as spaced out at the top is to vary them by 1.1% each step:

    {1, 2.1, 3.2, 4.3, 5.4, 6.5, 7.6, 8.7, 9.8, 10.9, 12, 13.1, 15.4}

    Hope this helps.


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